Secondary Objective Function Research Articles

An Analytical Framework for the Multiplicative Best‐Worst Method

ABSTRACTThe Best‐Worst Method (BWM) is a well‐known Multi‐Criteria Decision‐Making (MCDM) method. Numerous models of BWM have been developed by integrating various distance functions with the core principles of the method. This article addresses the multiplicative model of BWM. We first formulate an optimization model that is equivalent to the existing multiplicative model. This model provides a robust foundation for deriving analytical forms of optimal interval‐weights, the Consistency Index (CI) and the Consistency Ratio (CR). The proposed approach eliminates the need for optimization software, enhancing its ease of implementation and time efficiency. Moreover, the derived analytical form of CR enables it to function as an input‐based consistency measure. We then introduce a secondary objective function to select the best optimal weight set from the collection of all optimal weight sets. Finally, we present numerical examples to illustrate the proposed approach.

PC-ILP: A Fast and Intuitive Method to Place Electric Vehicle Charging Stations in Smart Cities

The widespread use of electric vehicles necessitates meticulous planning for the placement of charging stations (CSs) in already crowded cities so that they can efficiently meet the charging demand while adhering to various real-world constraints such as the total budget, queuing time, electrical regulations, etc. Many classical and metaheuristic-based approaches provide good solutions, but they are not intuitive, and they do not scale well for large cities and complex constraints. Many classical solution techniques often require prohibitive amounts of memory and their solutions are not easily explainable. We analyzed the layouts of the 50 most populous cities of the world and observed that any city can be represented as a composition of five basic primitive shapes (stretched to different extents). Based on this insight, we use results from classical topology to design a new charging station placement algorithm. The first step is a topological clustering algorithm to partition a large city into small clusters and then use precomputed solutions for each basic shape to arrive at a solution for each cluster. These cluster-level solutions are very intuitive and explainable. Then, the next step is to combine the small solutions to arrive at a full solution to the problem. Here, we use a surrogate function and repair-based technique to fix any resultant constraint violations (after all the solutions are combined). The third step is optional, where we show that the second step can be extended to incorporate complex constraints and secondary objective functions. Along with creating a full software suite, we perform an extensive evaluation of the top 50 cities and demonstrate that our method is not only 30 times faster but its solution quality is also 36.62% better than the gold standard in this area—an integer linear programming (ILP) approach with a practical timeout limit.

An analytical framework for the best–worst method

Since the development of the best–worst method (BWM) in 2015, it has become a popular research focus in multi-criteria decision-making. The original optimization problem of the BWM is a nonlinear min–max model that can lead to multiple optimal solutions, while the linear model of the BWM produces a unique solution. The two models need to be solved by optimization software packages. In addition, although the linear model of the BWM can obtain a unique solution, it produces different feasible regions than the nonlinear model of the BWM, and it changes the objective function. This study aims to solve the nonlinear model of the BWM mathematically to obtain the analytical forms of the optimal solutions. First, we transform the original nonlinear model of BWM into an equivalent optimization model driven by the optimally modified comparison vectors. The equivalent BWM provides a solid basis for computing the analytical solutions. Second, for not-fully consistent pairwise comparison systems, we strictly prove that there is only one unique optimal solution with three criteria, and there might be multiple optimal solutions with more than three criteria. We further develop the analytical forms of these unique and multiple optimal solutions and the optimal interval weights. Third, we develop a secondary objective function to select a unique solution for the BWM. The secondary objective function retains all the characteristics of the original nonlinear model of the BWM, and we find the unique solution analytically. Finally, some numerical examples are examined, and a comparative analysis is performed to demonstrate the effectiveness of our analytical solution approach.

Null-Space Minimization of Center of Gravity Displacementof a Redundant Aerial Manipulator

Displacements of the base during trajectory tracking are a common issue in the control of aerial manipulators. These are caused by reaction torques transferred to the base due to the manipulator motion and, in particular, to the motion of its center of gravity. We present a novel approach to reduce base displacements of a kinematically redundant aerial manipulator by using null-space projection in the inverse kinematic control. A secondary objective function minimizes the horizontal displacement of the manipulator center of gravity. We test this algorithm on different trajectories for both three and four degrees of freedom (DOF) manipulators in a simulation environment. The results comparing our algorithm with inverse kinematic control without the null-space projection show up to an 80% reduction in the end-effector position error and an average of about 56% reduction in maximum base displacement. The simulation implementation also runs faster than in real-time in our code implementation. We provide a workspace analysis based on multiple stopping criteria such as excessive base displacement, joint velocities and end-effector position error for the 3 and 4 DOF manipulators. As expected, the 4 DOF manipulator has a larger workspace.

Creating Shared Value and Strategic Corporate Social Responsibility through Outsourcing within Supply Chain Management

One way to develop local clusters is to strengthen those clusters by using outsourcing to conduct strategic social responsibility, or in other words, to create shared value, which is a win-win strategy for the buyer, supplier, and society and the best and most viable alternative to traditional corporate social responsibilities. In the leading research, a model for decision-making within the supply chain has been developed for purchasing based on shared value creation, long-term relationship management, and purchasing strategies. The research consists of two strategic mathematical models, using goal programming, and then is solved by a meta-heuristic algorithm. Potential outsourcing companies are assessed and then clustered according to their geographic locations in the decision-making process. One (or several) cluster(s) was selected among clusters based on knowledge and relationship criteria. Besides, in the primary mathematical model, the orders in different periods and the selection of suppliers are determined. In this model, in addition to optimizing the cost, the dispersion of purchases from suppliers is maximized to increase relationships and strengthen all members of the cluster. Maximizing the distribution by converting a secondary objective function to goal-programming variables transforms the multi-objective model into a single-objective model. In addition to economic benefits for buyers and suppliers, this purchasing plan concentrates on strengthening the local industrial cluster, fostering employment and ease of recruitment for human resources, accessing more infrastructures and technical support facilities, developing an education system in the region, and assisting knowledge-based enterprises with development.

Unbounded Serial-Batching Scheduling on Hierarchical Optimization

The paper considers a serial-batching scheduling problem on hierarchical optimization with two regular maximum costs, where hierarchical optimization means the primary objective function is minimized, and keeping the minimum value of the primary objective function, the secondary objective function is also minimized. In serial-batching machine environment, the machine processes the jobs in batch, and the jobs in the identical batch are processed by entering into the machine together and leaving the machine together. The time taken to process a batch amounts to the total processing time of the jobs in the batch. Moreover, a fixed switching time s is inserted when a machine begins to process a new batch. We only study the unbounded model, i.e., the batch capacity is unbounded. We give an algorithm that can solve the hierarchical optimization problem in $$O(n^4)$$ time, where n denotes the number of jobs.

Research on Task Assignment Model and Algorithm of Online Picking System Based on Multi-objective

In view of the task assignment problem of the picking station in the online Parts-to-Picker Picking System, a mathematical model was established that targeting the minimum number of total rounds of the bins and the equalization of the picking of each station, constrained by the uniqueness of the order attribution and the maximum amount of orders processed by the workstation. Then, the solution flow and method of multi-objective function problem based on Genetic Algorithm are designed. The algorithm constrains the secondary objective function and defines the rules of the objective function constraint. Finally, an empirical study on the actual Parts-to-Picker System of an e-commerce enterprise is carried out. The results show that the algorithm can effectively reduce the two objective function values and verify the effectiveness of the algorithm.

A novel method for wind farm layout optimization based on wind turbine selection

A novel method was developed to detect the optimal onshore wind farm layout driven by the characteristics of all commercially-available wind turbines. A huge number of turbine combinations (577) was processed, resulting in 22,721 generated layouts. Various assumptions and constraints were considered, mostly derived from the literature, including site features, wind conditions, and layout design. For the latter, an irregularly staggered turbine array configuration was assumed. Wake effects were simulated through the Jensen’s model, while a typical turbine thrust coefficient curve as a function of wind speed was originally developed. A detailed cost model was used, with levelized cost of energy selected as primary and capacity factor as secondary objective function. The self-organizing maps were used to address a thorough analysis, proving to be a powerful means to straightforwardly achieve a comprehensive pattern of wind farm layout optimization.In general, the two optimization functions basically match, while for higher wind potential sites, increasing capacity factor did not necessarily result in decreasing levelized cost of energy. The latter may be minimised by reducing the total number of turbines or the overall wind farm capacity, as well as maximising rotor diameters or minimising rated wind speeds; increasing rated power or hub height is only beneficial for mid-potential sites. The mere maximisation of wind farm energy production is a misleading target, as corresponding to mid-to-high values of levelized cost of energy. In contrast to previous studies, the use of turbines with different rated power, rotor diameter or hub height should be avoided.

A New Model for the Secondary Goal in DEA

The purpose of the current paper is to propose a new model for the secondary goal in DEA by introducing secondary objective function. The proposed new model minimizes the average of the absolute deviations of data points from their median. Similar problem is studied in a related model by Liang et al. (2008), which minimizes the average of the absolute deviations of data points from their mean. By using two well known data sets, which are also used by Liang et al.(2008), and Greene (1990) we compare the results of the proposed new model and several other models.

Relationship between Firm Efficiency and Stock Price Performance

Modern investment theory has empirically proved that stock returns can be explained by several factors such as market risk, firm size, and book-to-market ratio. Other unknown factors affecting stock returns are also believed to still exist yet to be found. We believe that one of such factors is the operational efficiency of firms in transforming inputs to outputs, considering the fact that operations is a fundamental and primary function of any type of businesses. To support this belief, this study intends to empirically study the relationship between firm efficiency and stock price performance. Firm efficiency is measured using data envelopment analysis (DEA) with inputs and outputs obtained from financial statements. We employ cross-efficiency evaluation to enhance the discrimination power of DEA with a secondary objective function of aggressive formulation. Using the CAPM-based performance regression model, we test the performance of equally weighted portfolios of different sizes selected based upon DEA cross-efficiency scores along with a buy & hold trading strategy. For the empirical test, we collect financial data of domestic firms listed in KOSPI over the period of 2000~2016 from well-known financial databases. As a result, we find that the porfolios with highly efficient firms included outperform the benchmark market portfolio after controlling for the market risk, which indicates that firm efficiency plays a important role in explaining stock returns.

Cross‐efficiency evaluation method based on the conservative point of view

AbstractTraditional cross‐efficiency evaluation models have ignored the problem that large differences may exist among cross‐efficiencies, which may make decision making units (DMUs) unwilling to accept cross‐efficiency evaluation results. Aimed at solving this problem, this paper proposes an altruism cross‐efficiency model based on the conservative point of view. Compared with other cross‐efficiency evaluation models, the proposed model mainly exhibits the following advantages. First, the proposed model no longer guarantees the DMU's self‐evaluation efficiency, allowing the self‐evaluation efficiency to adaptively change. Therefore, the peer‐evaluation process is flexible and adaptable to the actual situation. Second, the proposed model abandons the traditional max‐average secondary objective function and proposes to use a max–min objective function. Thus, the proposed model can maximize the peer‐efficiency of the worst performing peer‐DMU, achieving the effects of reducing the gaps among cross‐efficiencies. Third, the cross‐efficiency evaluation model is based on the conservative point of view, which helps the DMU to distinguish potential competitors among peer‐DMUs. Lastly, to solve the non‐linear model proposed in this paper, the algorithm is designed to describe how to solve the model. The case of a flexible manufacturing system is used to show the appropriateness of the suggested model and algorithm.

"On the Two Machine Permutation Flowshop Scheduling Problems Using Variable Neighborhood Search Algorithms "

"This paper considers the two machine permutation flow shop scheduling problem PFSSP. We study some special cases of the problem 2||( , Σ , , Σ ) (P). In the first one, the special cases to find the exact non dominated solutions for problem () are based on the relation between processing times , and the due dates on machine A and machine B, ( = 1,2, … , ). In the second one, the special cases are based on the lexicographical order minimization to solve three problems, in each one the primary objective function is maximum completion time ( ), while the secondary objective function is one of the objective functions: total completion times (Σ ), maximum tardiness ( ) and total tardiness (Σ ), the resulting problems are (P1), (P2), (P3). We propose three algorithms: Variable neighborhood simulated annealing(VNSA), Variable neighborhood descent method(VNDM) and Variable neighborhood with modefied NEH algorithm(VNMNEH). The comparison results show the efficiency of the proposed algorithms. All algorithmes were coded in Matlab programe."

Multi-stage generation and transmission coordinated planning method with a modified fireworks algorithm facing to energy internet

The integration of energy internet (EI) and electric power system has a profound influence on electric power system planning. Therefore, this paper mainly resolves the EI-oriented multi-stage generation and transmission coordinated planning problem with a modified fireworks algorithm. Firstly, we combined with Stackelberg game theory and focused on subject game theory, system low carbon characteristic and renewable energy generation, a multi-stage generation and transmission coordinated planning model for electric power system is proposed. The model has a main objective function of maximizing the social welfare and two secondary objective functions of maximizing individual profits in the operation stage and maximizing overall profits of generation companies in the planning stage. Then, a modified fireworks algorithm (MFWA) is proposed with good expandability, convergence and adaptability to solve the problem. The MFWA is modified by introducing Gaussian Mutation to improve global search capability. The basic coding method and solution procedure of the MFWA are depicted. Finally, an improved IEEE-24 bus system is used to test the proposed planning method and MFWA. By comparing three scenarios and analysing results, we can verify the validity and feasibility of the proposed method.

A multistage heuristic for storage and retrieval problems in a warehouse with random storage

AbstractThe warehouse is one of the essential components of logistics and supply chains. The efficiency of the whole chain is affected by the performance of warehouse operations and, more particularly, the storage and retrieval of goods. This paper considers a storage and retrieval problem in a real warehouse with random storage and different types of forklifts, depending on the locations they can access. The problem deals with selecting locations to store/retrieve a predefined set of pallets, assigning an adequately skilled forklift to each operation and determining the order in which each forklift will perform its operations so that the total employed time is minimized. The problem is solved heuristically by decomposing it into three subproblems, each one handling one of the three key decisions of the problem, and taking into account congestion considerations. The paper also studies two modifications of the problem, adding secondary objective functions. Computational results compare the effectiveness of the proposed algorithms for the different problems in a stochastic environment via simulation.

Control of the seven-degree-of-freedom upper limb exoskeleton for an improved human-robot interface

This study analyzes a practical scheme for controlling an exoskeleton robot with seven degrees of freedom (DOFs) that supports natural movements of the human arm. A redundant upper limb exoskeleton robot with seven DOFs is mechanically coupled to the human body such that it becomes a natural extension of the body. If the exoskeleton robot follows the movement of the human body synchronously, the energy exchange between the human and the robot will be reduced significantly. In order to achieve this, the redundancy of the human arm, which is represented by the swivel angle, should be resolved using appropriate constraints and applied to the robot. In a redundant 7-DOF upper limb exoskeleton, the pseudoinverse of the Jacobian with secondary objective functions is widely used to resolve the redundancy that defines the desired joint angles. A secondary objective function requires the desired joint angles for the movement of the human arm, and the angles are estimated by maximizing the projection of the longest principle axis of the manipulability ellipsoid for the human arm onto the virtual destination toward the head region. Then, they are fed into the muscle model with a relative damping to achieve more realistic robot-arm movements. Various natural arm movements are recorded using a motion capture system, and the actual swivel-angle is compared to that estimated using the proposed swivel angle estimation algorithm. The results indicate that the proposed algorithm provides a precise reference for estimating the desired joint angle with an error less than 5◦.

Determination of lower and upper bounds of predicted production from history-matched models

We present a method to determine lower and upper bounds to the predicted production or any other economic objective from history-matched reservoir models. The method consists of two steps: 1) performing a traditional computer-assisted history match of a reservoir model with the objective to minimize the mismatch between predicted and observed production data through adjusting the grid block permeability values of the model. 2) performing two optimization exercises to minimize and maximize an economic objective over the remaining field life, for a fixed production strategy, by manipulating the same grid block permeabilities, however without significantly changing the mismatch obtained under step 1. This is accomplished through a hierarchical optimization procedure that limits the solution space of a secondary optimization problem to the (approximate) null space of the primary optimization problem. We applied this procedure to two different reservoir models. We performed a history match based on synthetic data, starting from a uniform prior and using a gradient-based minimization procedure. After history matching, minimization and maximization of the net present value (NPV), using a fixed control strategy, were executed as secondary optimization problems by changing the model parameters while staying close to the null space of the primary optimization problem. In other words, we optimized the secondary objective functions, while requiring that optimality of the primary objective (a good history match) was preserved. This method therefore provides a way to quantify the economic consequences of the well-known problem that history matching is a strongly ill-posed problem. We also investigated how this method can be used as a means to assess the cost-effectiveness of acquiring different data types to reduce the uncertainty in the expected NPV.

Aeroelastic Shape Optimization of a Flapping Wing

This paper presents the theory and results for the shape and structural optimization of a platelike flapping wing. The aeroelastic system is analyzed by coupling an unsteady vortex lattice aerodynamics model with a plate finite element model. The assumptions in the aerodynamic model allow the system of equations to be calculated with the inversion of a single matrix, greatly reducing the computational cost. The design variables are the shape parameters from the modified Zimmerman method and the polynomial coefficients that describe the wing thickness. The wing shape and structure are optimized using two multiobjective optimization formulations. The first optimization minimizes the input power while maximizing the cycle-averaged thrust. The input power is the secondary objective function and is treated as a nonlinear constraint, whereas the cycle-averaged thrust is the primary objective function. A second multiobjective formulation that treats wing mass as the secondary objective function is also performed. The power-thrust-optimal wing designs minimize the contribution of the wing deformation to the input power over the flapping cycle. The mass-optimal shapes maximize the wingspan and tailor the wing thickness such that the wing deformation adds to the thrust. It is shown that, although thrust benefits from added wing deformation, it also adds to the power required to flap the wing.

Dual objective motion planning subject to state constraints

SUMMARYThis paper presents a novel motion planning approach inspired by the Dynamic Programming (DP) applicable to multi degree of freedom robots (mobile or stationary) and autonomous vehicles. The proposed discrete–time algorithm enables a robot to reach its destination through an arbitrary obstacle field in the fewest number of time steps possible while minimizing a secondary objective function. Furthermore, the resulting optimal trajectory is guaranteed to be globally optimal while incorporating state constraints such as velocity, acceleration, and jerk limits. The optimal trajectories furnished by the algorithm may be further updated in real time to accommodate changes in the obstacle field and/or cost function. The algorithm is proven to terminate in a finite number of steps without its computational complexity increasing with the type or number of obstacles. The effectiveness of the global and replanning algorithms are demonstrated on a planar mobile robot with three degrees of freedom subject to velocity and acceleration limits. The computational complexity of the two algorithms are also compared to that of an A*–type search.

Sensitivity‐based optimal shape design of induction‐heating devices

A design of experiment (DOE) strategy applied to multi-objective optimisation is proposed in order to evaluate the influence of design variables variations to optimised quantities. A secondary objective function is the sensitivity of a primary objective function to design variable variations evaluated by means of DOE strategy. The optimisation problem includes also a third objective function that considers device constraint because of technological limitations on power generator. The proposed case study deals with the design of an electromagnetic device that will be used to carry out laboratory experiments on magneto-fluid hyperthermia, that is, a clinic treatment for cancer cure. The induction system is designed to apply a controlled time-varying magnetic field to biological cells, cultured in Petri dish and mixed with magnetic nanoparticles. This study presents an original cost-effective method of multi-objective design optimisation taking into account design uncertainties.

Choosing effective dates from multiple optima in Technology Forecasting using Data Envelopment Analysis (TFDEA)

Technology Forecasting using Data Envelopment Analysis (TFDEA) provides an effective means to forecast technological capability over time without the burden of fixed a priori weighting schemes. However, there are situations where result reproduction can be a challenge as first pointed out in a previous Technological Forecasting and Social Change article [11]. When using a commonly used extension of TFDEA, there are circumstances where multiple optimal solutions can complicate analysis. This paper addresses this issue through extending the TFDEA model in a manner consistent with common Data Envelopment Analysis (DEA) techniques. The extension is then demonstrated using datasets from previous publications on fighter jet and commercial airplane technology where the issue of multiple optima has been observed. The results indicate that traditional TFDEA may generate varying forecasts depending on the software used, which can be dealt with by introducing a secondary objective function. Therefore, researchers should explicitly state which secondary objective function they are using for the TFDEA applications.